How to animate dynamic scaling and z offset in Array modifier
Using an empty as Object offset to control the array modifier will give you a lot of options.
If you play with the reltive offset you can get things like this:



Changing the distance and size of the empty you'll control the relative distance of the cloned items in the array.

You can then add keyframes:



By adding a lattice on the scene and a lattice modifier to the plane to the stack you can further control the deformation (as in the opening of the video used as reference on the question: